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Theorem caofdi 6355
Description: Transfer a distributive law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypotheses
Ref Expression
caofdi.1  |-  ( ph  ->  A  e.  V )
caofdi.2  |-  ( ph  ->  F : A --> K )
caofdi.3  |-  ( ph  ->  G : A --> S )
caofdi.4  |-  ( ph  ->  H : A --> S )
caofdi.5  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  S  /\  z  e.  S ) )  -> 
( x T ( y R z ) )  =  ( ( x T y ) O ( x T z ) ) )
Assertion
Ref Expression
caofdi  |-  ( ph  ->  ( F  oF T ( G  oF R H ) )  =  ( ( F  oF T G )  oF O ( F  oF T H ) ) )
Distinct variable groups:    x, y,
z, A    x, F, y, z    x, G, y, z    ph, x, y, z   
x, H, y, z   
x, K, y, z   
x, O, y, z   
x, R, y, z   
x, S, y, z   
x, T, y, z
Allowed substitution hints:    V( x, y, z)

Proof of Theorem caofdi
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 caofdi.5 . . . . 5  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  S  /\  z  e.  S ) )  -> 
( x T ( y R z ) )  =  ( ( x T y ) O ( x T z ) ) )
21adantlr 709 . . . 4  |-  ( ( ( ph  /\  w  e.  A )  /\  (
x  e.  K  /\  y  e.  S  /\  z  e.  S )
)  ->  ( x T ( y R z ) )  =  ( ( x T y ) O ( x T z ) ) )
3 caofdi.2 . . . . 5  |-  ( ph  ->  F : A --> K )
43ffvelrnda 5840 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  K )
5 caofdi.3 . . . . 5  |-  ( ph  ->  G : A --> S )
65ffvelrnda 5840 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
7 caofdi.4 . . . . 5  |-  ( ph  ->  H : A --> S )
87ffvelrnda 5840 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( H `  w )  e.  S )
92, 4, 6, 8caovdid 6277 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) T ( ( G `  w ) R ( H `  w ) ) )  =  ( ( ( F `  w ) T ( G `  w ) ) O ( ( F `  w ) T ( H `  w ) ) ) )
109mpteq2dva 4375 . 2  |-  ( ph  ->  ( w  e.  A  |->  ( ( F `  w ) T ( ( G `  w
) R ( H `
 w ) ) ) )  =  ( w  e.  A  |->  ( ( ( F `  w ) T ( G `  w ) ) O ( ( F `  w ) T ( H `  w ) ) ) ) )
11 caofdi.1 . . 3  |-  ( ph  ->  A  e.  V )
12 ovex 6115 . . . 4  |-  ( ( G `  w ) R ( H `  w ) )  e. 
_V
1312a1i 11 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) R ( H `
 w ) )  e.  _V )
143feqmptd 5741 . . 3  |-  ( ph  ->  F  =  ( w  e.  A  |->  ( F `
 w ) ) )
155feqmptd 5741 . . . 4  |-  ( ph  ->  G  =  ( w  e.  A  |->  ( G `
 w ) ) )
167feqmptd 5741 . . . 4  |-  ( ph  ->  H  =  ( w  e.  A  |->  ( H `
 w ) ) )
1711, 6, 8, 15, 16offval2 6335 . . 3  |-  ( ph  ->  ( G  oF R H )  =  ( w  e.  A  |->  ( ( G `  w ) R ( H `  w ) ) ) )
1811, 4, 13, 14, 17offval2 6335 . 2  |-  ( ph  ->  ( F  oF T ( G  oF R H ) )  =  ( w  e.  A  |->  ( ( F `  w ) T ( ( G `
 w ) R ( H `  w
) ) ) ) )
19 ovex 6115 . . . 4  |-  ( ( F `  w ) T ( G `  w ) )  e. 
_V
2019a1i 11 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) T ( G `
 w ) )  e.  _V )
21 ovex 6115 . . . 4  |-  ( ( F `  w ) T ( H `  w ) )  e. 
_V
2221a1i 11 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) T ( H `
 w ) )  e.  _V )
2311, 4, 6, 14, 15offval2 6335 . . 3  |-  ( ph  ->  ( F  oF T G )  =  ( w  e.  A  |->  ( ( F `  w ) T ( G `  w ) ) ) )
2411, 4, 8, 14, 16offval2 6335 . . 3  |-  ( ph  ->  ( F  oF T H )  =  ( w  e.  A  |->  ( ( F `  w ) T ( H `  w ) ) ) )
2511, 20, 22, 23, 24offval2 6335 . 2  |-  ( ph  ->  ( ( F  oF T G )  oF O ( F  oF T H ) )  =  ( w  e.  A  |->  ( ( ( F `
 w ) T ( G `  w
) ) O ( ( F `  w
) T ( H `
 w ) ) ) ) )
2610, 18, 253eqtr4d 2483 1  |-  ( ph  ->  ( F  oF T ( G  oF R H ) )  =  ( ( F  oF T G )  oF O ( F  oF T H ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   _Vcvv 2970    e. cmpt 4347   -->wf 5411   ` cfv 5415  (class class class)co 6090    oFcof 6317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319
This theorem is referenced by:  psrlmod  17462  plydivlem4  21705  plydiveu  21707  quotcan  21718  basellem9  22369  mendlmod  29459  lflvsdi2  32412
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